I'm doing a little research on elementary number theory, and found this question online. I thought it should be easy for me to prove, but that seemed not to be true:
Let $p, q$ be primes.
Prove: gcd($x, pq$) = 1 $\Rightarrow$ gcd($x, p$) = 1 and gcd($x, q$) = 1
I tried rewriting this to: $1 = ax + bpq$ for some integers $a, b$ (using the extended euclidean algorithm) but I got stuck there, so I think that isn't the right approach.